Finite Group DatabasebetaFinite Group Databasebeta| order | 8001 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #11765917⋊C7 |
| derived, Sylow | C127 ⋊ C7×C9 |
| Hall | #11765917 ⋊ C7 |
| Hall | C127⋊C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | (C127⋊C3)⋊C7 |
| commutator, hypocenter | C127 | C7×C9 |
| Fitting, socle | C3×C127 | C3×C7 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C3 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
3,7,127,3|a^3=d,c^a=c^107,c^b=c^481 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11765917⋊C7, (#11765917⋊C2)⋊C7, C22⋊(#11765917⋊C7) |
| maximal normal subgroup of | C2×#11765917⋊C7, (#11765917⋊C2)⋊C7 |
| maximal quotient of | C2×#11765917⋊C7, C22⋊(#11765917⋊C7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8001, 2);